QR decomposition

About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.

The Forward Search for Very Large Datasets

Abstract: The identification of atypical observations and the immunization of data analysis against both outliers and failures of modeling are important aspects of modern statistics. The forward search is a graphics rich approach that leads to the formal detection of outliers and to the detection of model inadequacy combined with suggestions for model enhancement. The key idea is to monitor quantities of interest, such as parameter estimates and test statistics, as the model is fitted to data subsets of increasing size. In this paper we propose some computational improvements of the forward search algorithm and we provide a recursive implementation of the procedure which exploits the information of the previous step. The output is a set of efficient routines for fast updating of the model parameter estimates, which do not require any data sorting, and fast computation of likelihood contributions, which do not require matrix inversion or qr decomposition. It is shown that the new algorithms enable a reduction of the computation time by more than 80%. Furthemore, the running time now increases almost linearly with the sample size. All the routines described in this paper are included in the FSDA toolbox for MATLAB which is freely downloadable from the internet.

A robust watermarking algorithm based on QR factorization and DCT using quantization index modulation technique

Abstract: We propose a robust digital watermarking algorithm for copyright protection. A stable feature is obtained by utilizing QR factorization and discrete cosine transform (DCT) techniques, and a meaningful watermark image is embedded into an image by modifying the stable feature with a quantization index modulation (QIM) method. The combination of QR factorization, DCT, and QIM techniques guarantees the robustness of the algorithm. Furthermore, an embedding location selection method is exploited to select blocks with small modifications as the embedding locations. This can minimize the embedding distortion and greatly improve the imperceptibility of our scheme. Several standard images were tested and the experimental results were compared with those of other published schemes. The results demonstrate that our proposed scheme can achieve not only better imperceptibility, but also stronger robustness against common signal processing operations and lossy compressions, such as filtering, noise addition, scaling, sharpening, rotation, cropping, and JPEG/JPEG2000 compression.

Hybrid algorithm for fast Toeplitz orthogonalization

Abstract: New techniques for fast Toeplitz QR decomposition are presented The methods are based on the shift invariance property of a Toeplitz matrix The numerical properties of the algorithms are discussed and some comparisons are made with two other fast Toeplitz orthogonalization methods

Computing 3SLS Solutions of Simultaneous Equation Models with a Possible Singular Variance–Convariance Matrix

Abstract: Algorithms for computing the three-stage least squares (3SLS) estimator usually require the disturbance convariance matrix to be non-singular. However, the solution of a reformulated simultaneous equation model (SEM) results into the redundancy of this condition. Having as a basic tool the QR decomposition, the 3SLS estimator, its dispersion matrix and methods for estimating the singular disturbance covariance matrix and derived. Expressions revealing linear combinations between the observations which become redundant have also been presented. Algorithms for computing the 3SLS estimator after the SEM have been modified by deleting or adding new observations or variables are found not to be very efficient, due to the necessity of removing the endogeneity of the new data or by re-estimating the disturbance covariance matrix. Three methods have been described for solving SEMs subject to separable linear equalities constraints. The first method considers the constraints as additional precise observations while the other two methods reparameterized the constraints to solve reduced unconstrained SEMs. Method for computing the main matrix factorizations illustrate the basic principles to be adopted for solving SEMs on serial or parallel computers.